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\noindent\parbox{2.95cm}{\includegraphics*[keepaspectratio=true,scale=0.125]{AFA.jpg}}
\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
\qquad Ann. Funct. Anal. 3 (2012), no. 1, 92--99\\
{\footnotesize \qquad \textsc{\textbf{$\mathscr{A}$}nnals of
\textbf{$\mathscr{F}$}unctional
\textbf{$\mathscr{A}$}nalysis}\\
\qquad ISSN: 2008-8752 (electronic)\\
\qquad URL:
\textcolor[rgb]{0.00,0.00,0.99}{www.emis.de/journals/AFA/}
}\\[.5in]}
\title[Approximation of Analytic Functions by Special Functions]
{Approximation of Analytic Functions by Special Functions}
\author[S.-M. Jung]{Soon-Mo Jung}
\address{Mathematics Section,
College of Science and Technology,
Hongik University,
339--701 Jochiwon, Republic of Korea.}
\email{\textcolor[rgb]{0.00,0.00,0.84}
{smjung@hongik.ac.kr;smjung57@yahoo.co.kr}}
\dedicatory{{\rm Communicated by J. Chmieli\'nski}}
\subjclass[2010]{Primary 34A05; Secondary 39B82, 26D10, 34A40,
46N20.}
\keywords{Hyers--Ulam stability, power series method, special
function, approximation.}
\date{Received: 9 January 2012; Accepted: 30 January 2012.}
\begin{abstract}
We survey the recent results concerning the applications of power
series method to the study of Hyers--Ulam stability of differential
equations.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:1}
Differential equations have been studied for more than $300$
years since the $17$th century when the concepts of
differentiation and integration were formulated by Newton and
Leibniz.
By use of differential equations, we can explain many natural
phenomena: gravity, projectiles, wave, vibration, nuclear
physics and so on.
Let us consider a closed system which can be explained by the first
order linear differential equation, namely, $y'(t) = \lambda y(t)$.
The past, present, and future of this system are completely
determined if we know the general solution and an initial condition
of that differential equation. So we can say that this system is
`predictable.' Sometimes, because of the disturbances (or noises) of
the outside, the system may not be determined by $y'(t) = \lambda
y(t)$ but can only be explained by an inequality like $| y'(t) -
\lambda y(t) | \leq \varepsilon$. Then it is impossible to predict
the exact future of the disturbed system.
Even though the system is not predictable exactly because of outside
disturbances, we say the differential equation $y'(t) = \lambda
y(t)$ has the Hyers--Ulam stability if the `real' future of the
system follows the solution of $y'(t) = \lambda y(t)$ with a bounded
error (see \cite{czerw,forti,4,5,hyersrassias,jung,10,12} for the
exact definition of Hyers--Ulam stability). But if the error bound
is `too big,' we say that differential equation $y'(t) = \lambda
y(t)$ does not have the Hyers--Ulam stability. Resonance is the
case.
There is another way to explain the Hyers--Ulam stability. Usually
the experiment (or the observed) data do not exactly coincide with
theoretical ones. We may express natural phenomena by use of
equations but because of the errors due to measurement or observance
the actual experiment data can almost always be a little bit off the
expectations. If we would use inequalities instead of equalities to
explain natural phenomena, then these errors could be absorbed into
the solutions of inequalities, i.e., those errors would be no more
errors.
Considering this point of view, the Hyers--Ulam stability (of
differential equations) is fundamental. (The Hyers--Ulam stability
is not same as the concept of the stability of differential
equations which has been studied by many mathematicians from long
time ago).
The Hyers--Ulam stability of functional equations has been studied
for more than $70$ years. But the history of the Hyers--Ulam
stability of differential equations is less than $20$ years. For
example, Ob\a{l}oza seems to be the first author who has
investigated in $1993$ the Hyers--Ulam stability of linear
differential equations (see \cite{ob1,ob2}). Thereafter, Alsina and
Ger published their paper \cite{1}, which handles the Hyers--Ulam
stability of the linear differential equation $y'(x) = y(x)$: If a
differentiable function $y(x)$ is a solution of the inequality $|
y'(x) - y(x) | \leq \varepsilon$ for any $x \in (a, \infty)$, then
there exists a constant $c$ such that $| y(x) - ce^x | \leq
3\varepsilon$ for all $x \in (a, \infty)$. We know that the general
solution of the linear differential equation $y'(x) = y(x)$ is $y(x)
= ce^x$, where $c$ is a constant.
Therefore, we say that the differential equation $y'(x) = y(x)$ has
the Hyers--Ulam stability. If we can get a similar result with a
control function $\varphi(x)$ in place of $\varepsilon$, we say that
the differential equation $y'(x) = y(x)$ has the Hyers--Ulam-Rassias
stability.
In $2001$ and $2002$, Miura et al. \cite{11} expanded Alsina and
Ger's result by proving that the differential equation $y'(x) =
\lambda y(x)$ has the Hyers--Ulam stability. The author wrote a
paper with Miura and Takahasi which expanded the result of
Hyers--Ulam stability of that differential equation. To be more
precise, we may choose a constant $c$ such that the solution of the
inequality $| y'(x) - \lambda y(x) | \leq \varphi(x)$ is not too far
away from $ce^{\lambda x}$ in the sense of upper norm (see
\cite{18}).
As above, still not many papers were published about the Hyers--Ulam
stability of differential equations. The author has been studying
this subject since Alsina and Ger's paper, and published several
papers (ref. \cite{jung1,jung0,jung3,j15,j13}).
In this paper, we will survey the recent results concerning the
applications of power series method to the study of Hyers--Ulam
stability of differential equations.
\section{Power series method}
\label{sec:2}
In $2007$, using the power series method, the author \cite{jung4}
proved the Hyers--Ulam stability of the Legendre's differential
equation
$$
\big( 1 - x^2 \big) y''(x) - 2x y'(x) + p(p + 1) y(x) = 0,
$$
where $p$ is a real number, whose solutions are called Legendre
functions.
The Legendre's differential equation plays a great role in
physics and engineering.
In particular, this equation is most useful for treating the
boundary value problems exhibiting spherical symmetry.
To the best of our knowledge, the author first applied the power
series method to the Hyers--Ulam stability problems. Here, we will
introduce a theorem from \cite{j9}:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Theorem 2.1 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{thm:Legendre}
Assume that $\rho$ and $\rho_0$ are positive constants with
$\rho < \rho_0$.
Let $y : (-\rho, \rho) \to \mathbb{C}$ be a function which can
be represented by a power series
$y(x) = \sum_{m=0}^\infty b_m x^m$ whose radius of convergence
is $\rho_0$.
Assume moreover that there exists a positive number $\rho_1$
for which the condition
$$
\rho_1 = \lim_{k\to\infty} \left| \frac{c_k}{c_{k+1}} \right|
> 0.
$$
is satisfied, where
$$
c_k = b_k - \frac{b_{k-2[k/2]}}{k!}
\prod_{j=1}^{[k/2]} (k - 2j - p)(k - 2j + p + 1)
$$
for all $k \in \{ 2, 3, \ldots \}$.
If $\rho < \min\{ 1, \rho_0, \rho_1 \}$, then there exist a
Legendre function $y_h : (-\rho, \rho) \to \mathbb{C}$ and a
constant $K > 0$ such that
$$
| y(x) - y_h(x) | \leq \frac{Kx^2}{1-x^2}
$$
for all $x \in (-\rho, \rho)$.
\end{theorem}
Thereafter, the author \cite{jung5} applied the power series
method to calculate an upper bound of error when we are
approximating analytic function with Airy function about the
origin, where an Airy function is a solution of the Airy's
differential equation
$$
y''(x) - x y(x) = 0
$$
which appears in many calculations in applied mathematics.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Theorem 2.2 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{thm:Airy}
Assume that $\rho$ and $\rho_0$ are positive constants with
$\rho < \min\{ \sqrt[3]{6}, \rho_0 \}$.
If a function $y : (-\rho, \rho) \to \mathbb{C}$ can be
represented by a power series
$y(x) = \sum_{m=0}^{\infty} b_m x^m$,
whose radius of convergence is $\rho_0$, then there exist an
Airy function $y_h : (-\rho, \rho) \to \mathbb{C}$ and a
constant $K > 0$ such that
$$
| y(x) - y_h(x) | \leq Kx^2
$$
for all $x \in (-\rho, \rho)$.
\end{theorem}
The Hermite's differential equation
$$
y''(x) - 2x y'(x) + 2\lambda y(x) = 0
$$
plays an important role in quantum mechanics, probability
theory, statistical mechanics, and in solutions of Laplace's
equation in parabolic coordinates.
Every solution of the Hermite's differential equation is called
an Hermite function.
Also, the author found an approximation formula of analytic
functions in terms of Hermite functions (ref. \cite{jung6}):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Theorem 2.3 %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{thm:Hermite}
Let $\lambda$ be a fixed real number not less than $1$.
Assume that $\rho$ and $\rho_0$ are positive constants with
$\rho < \rho_0$.
If a function $y : (-\rho, \rho) \to \mathbb{C}$ can be
represented by a power series
$y(x) = \sum_{m=0}^{\infty} b_m x^m$, whose radius of
convergence is $\rho_0$, then there exist an Hermite function
$y_h : (-\rho, \rho) \to \mathbb{C}$ and a constant $K > 0$
such that
$$
| y(x) - y_h(x) | \leq Kx^2 e^{x^2}
$$
for all $x \in (-\rho, \rho)$, where $K$ depends on $\lambda$,
$\rho$ and $y$.
\end{theorem}
The Kummer's differential equation
$$
x y''(x) + ( \beta - x ) y'(x) - \alpha y(x) = 0,
$$
which is also called the confluent hypergeometric differential
equation, appears frequently in practical problems and
applications.
The Kummer's differential equation has a regular singularity
at $x = 0$ and an irregular singularity at $\infty$.
We define $(\alpha)_m$ by $(\alpha)_0 = 1$ and
$(\alpha)_m = \alpha (\alpha + 1) (\alpha + 2) \cdots
(\alpha + m -1)$ for all $m \in \mathbb{N}$.
Moreover, we define
$$
I_\rho = \left\{\begin{array}{ll}
(-\rho, \rho)
& \mbox{(for $\beta < 1$)}, \vspace{1mm}\\
(-\rho, \rho) \backslash \{ 0 \}
& \mbox{(for $\beta > 1$)}
\end{array}
\right.
$$
for any $0 < \rho \leq \infty$.
For a given $\kappa \geq 0$, let us denote ${\mathcal K}_\kappa$
the set of all functions $y : I_\rho \to \mathbb{C}$ with the
properties:
\begin{itemize}
\item[$(K_1)$] $y(x)$ is represented by a power series
$\sum_{m=0}^\infty b_m x^m$ whose radius of
convergence is at least $\rho$;
\item[$(K_2)$] $\sum_{m=0}^\infty | a_m x^m | \leq
\kappa | \sum_{m=0}^\infty a_m x^m |$ for all
$x \in I_\rho$, where
$a_m = (m + \beta)(m + 1) b_{m+1} -
(m+\alpha) b_m$ for each $m \in \mathbb{N}_0$.
\end{itemize}
In the following theorem, a local Hyers--Ulam stability of the
Kummer's differential equation is investigated (see \cite{j16}):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Theorem 2.4 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{thm:Kummer}
Let $\alpha$ and $\beta$ be real constants such that
$\beta \not\in \mathbb{Z}$ and neither $\alpha$ nor
$1+\alpha-\beta$ is a nonpositive integer.
Suppose a function $y : I_\rho \to \mathbb{C}$ is representable
by a power series $\sum_{m=0}^\infty b_m x^m$ whose radius of
convergence is at least $\rho > 0$.
Assume that there exist nonnegative constants $\mu \neq 0$ and
$\nu$ satisfying the condition
$$
\left| \frac{(m-1)! (\beta)_m a_m}{(\alpha)_{m+1}} \right|
\leq \mu \left| \sum_{i=0}^{m-1}
\frac{i! (\beta)_i a_i}{(\alpha)_{i+1}}
\right|
\leq \nu \left| \frac{(m+1)! (\beta)_m a_m}{(\alpha)_{m+1}}
\right|
$$
for all $m \in \mathbb{N}_0$, where
$a_m = (m + \beta)(m + 1) b_{m+1} - (m + \alpha) b_m$.
$($Indeed, it is sufficient for the first inequality to hold
for all sufficiently large integers $m$.$)$
Let us define $\rho_0 = \min\{ \rho, 1/\mu \}$.
If $y \in {\mathcal K}_\kappa$ and it satisfies the differential
inequality
$$
\big| x y''(x) + ( \beta - x ) y'(x) - \alpha y(x) \big|
\leq \varepsilon
$$
for all $x \in I_{\rho_0}$ and for some $\varepsilon \geq 0$,
then there exists a solution $y_h : I_\infty \to \mathbb{C}$
of the Kummer's differential equation such that
$$
| y(x) - y_h(x) |
\leq \left\{
\begin{array}{ll}
{\displaystyle \frac{\nu}{\mu} \cdot
\frac{2\alpha - 1}{\alpha} \kappa \varepsilon}
& \mbox{$($for $\alpha > 1$$)$}, \vspace{3mm}\\
{\displaystyle \frac{\nu}{\mu}
\left[ \sum_{m=0}^{m_0-1}
\left| \left| \frac{m+1}{m+\alpha} \right| -
\left| \frac{m+2}{m+1+\alpha} \right|
\right| +
\frac{m_0+1}{m_0+\alpha}
\right]\! \kappa \varepsilon}
& \mbox{$($for $\alpha \leq 1$$)$}
\end{array}
\right.
$$
for any $x \in I_{\rho_0}$, where
$m_0 = max\{ 0, \lceil -\alpha \rceil \}$.
\end{theorem}
A function is called a Bessel function (of fractional order)
if it is a solution of the Bessel's differential equation
$$
x^2 y''(x) + x y'(x) + (x^2 - \nu^2 ) y(x) = 0,
$$
where $\nu$ is a positive nonintegral number.
The Bessel's differential equation plays a great role in
physics and engineering.
In particular, this equation is most useful for treating the
boundary value problems exhibiting cylindrical symmetries.
Let us define $I_\rho = (-\rho, \rho) \backslash \{ 0 \}$ for
a positive constant $\rho$.
For a given $\kappa \geq 0$, we denote by ${\mathcal B}_\kappa$
the set of all functions $y : I_\rho \to \mathbb{C}$ with the
properties:
\begin{itemize}
\item[$(B_1)$] $y(x)$ is expressible by a power series
$\sum_{m=0}^\infty b_m x^m$ whose radius of
convergence is at least $\rho$;
\item[$(B_2)$] $\sum_{m=0}^\infty | a_m x^m | \leq \kappa
| \sum_{m=0}^\infty a_m x^m |$ for any
$x \in I_\rho$, where
$a_m = b_{m-2} + (m^2 - \nu^2) b_m$ for all
$m \in \mathbb{N}_0$ and set
$b_{-2} = b_{-1} = 0$.
\end{itemize}
For the given positive nonintegral number $\nu$, define
$$
\begin{array}{l}
M_e(x) =
{\displaystyle \max\left\{ \prod_{j=i}^k
\frac{x^2}{| \nu^2 - (2j)^2 |} :
0 \leq i \leq k \leq \mu \right\}},
\vspace{3mm}\\
M_o(x) =
{\displaystyle \max\left\{ \prod_{j=i}^k
\frac{x^2}{| \nu^2 - (2j+1)^2 |} :
0 \leq i \leq k \leq \mu \right\}},
\vspace{3mm}\\
M(x) = \max\{ M_e(x),\, M_o(x),\, 1 \},
\end{array}
$$
where $\mu = [ \sqrt{\nu^2 + x^2}/2 ]$ and
$$
L_\nu = \sum_{m=0}^\infty \frac{1}{(m-\nu)^2} < \infty.
$$
We remark that $M(x) \to 1$ as $|x| \to 0$.
In the following theorem, a local Hyers--Ulam stability of the
Bessel's differential equation is investigated (see
\cite{jungbessel} and ref. \cite{kj}):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Theorem 2.5 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{thm:Bessel}
Let $\nu$ and $p$ be a positive nonintegral number and a
nonnegative integer with $p < \nu < p+1$, respectively.
Assume that a function $y \in {\mathcal B}_\kappa$ satisfies
the differential inequality
$$
\big| x^2 y''(x) + x y'(x) + (x^2 - \nu^2 ) y(x) \big|
\leq \varepsilon
$$
for all $x \in I_\rho$ and for some $\varepsilon \geq 0$.
If the sequence $\{ b_m \}$ satisfies the condition
$$
b_{m+2} = O(b_{m}) ~~\mbox{as}~~ m \to \infty
$$
with a Landau constant $C \geq 0$, then there exists a Bessel
function $y_h(x)$ such that
$$
| y(x) - y_h(x) | \leq \kappa L_\nu M(x) \varepsilon
$$
for any $x \in I_{\rho_0}$, where
$\rho_0 = \min\{ \rho, 1/\sqrt{C^\ast} \}$ and $C^\ast$ is a
positive number larger than $C$.
If $C$ is sufficiently small and $\rho$ is large, then
$$
M(x) \leq \max\left\{ \frac{|x|^{|x|+2}}{| \nu^2-p^2 |^{|x|/2+1}},\,
\frac{|x|^{|x|+2}}{| \nu^2-(p+1)^2 |^{|x|/2+1}}
\right\}
$$
for all sufficiently large $|x|$.
\end{theorem}
Each solution of the Laguerre's differential equation
$$
x y''(x) + (1-x) y'(x) + n y(x) = 0
$$
is called a Laguerre function.
The Laguerre's differential equation is a special case of the
Sturm-Liouville boundary problem.
This equation can deal with harmonic oscillator phenomena in
quantum mechanics.
For instance, the Laguerre's differential equation can describe
the state of electrons in the field of the Coulomb force.
Indeed, the solution of radial part of the Schr\"{o}dinger
equation for the electron in hydrogen atom consists of
functions expressed by Laguerre polynomials.
Let $\kappa \geq 0$ and $1 < \rho \leq \infty$ be constants.
We denote by ${\mathcal L}_\kappa$ the set of all functions
$y : [ 0, \rho ) \to \mathbb{C}$ with the following properties:
\begin{itemize}
\item[$(L_1)$] $y(x)$ is expressible by a power series
$\sum_{m=1}^\infty b_m x^m$ whose radius of
convergence is at least $\rho$;
\item[$(L_2)$] $\sum_{m=0}^\infty | a_m x^m | \leq \kappa
| \sum_{m=0}^\infty a_m x^m |$ for any
$x \in [ 0, 1 ]$, where
$a_m = (m+1)^2 b_{m+1} +(n-m) b_m$ for all
$m \in \mathbb{N}_0$ and set $b_0 = 0$.
\end{itemize}
The author applied the power series method to the study of
Hyers--Ulam stability of the Laguerre's differential equation (see
\cite{jung9}):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Theorem 2.6 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{thm:Laguerre}
Let $n$ be a nonnegative integer.
If a function $y \in {\mathcal L}_\kappa$ satisfies the
differential inequality
$$
\big| x y''(x) + (1-x) y'(x) + n y(x) \big| \leq \varepsilon
$$
for all $x \in [ 0, \rho )$ and for some $\varepsilon > 0$,
then there exists a Laguerre function
$y_h : [ 0, 1 ) \to \mathbb{C}$ such that
$$
| y(x) - y_h(x) | \leq \frac{\pi^2}{6} \kappa M \varepsilon x
$$
for all $x \in [ 0, 1 )$, where $M$ is defined by
$$
M = \left\{
\begin{array}{ll}
1 & ( n \in \{ 0, 1 \} ), \vspace{1mm} \\
{\displaystyle \max_{1 \leq k \leq \ell \leq [\sqrt{n}]}
\prod_{j=k}^\ell \frac{| n-j |}{j^2}}
& ( n \geq 2 ).
\end{array}
\right.
$$
\end{theorem}
Every solution of the Chebyshev's differential equation
$$
(1-x^2) y''(x) -x y'(x) + n^2 y(x) = 0
$$
is called a Chebyshev function.
The Chebyshev's differential equation has regular singular
points at $-1$, $1$, and $\infty$ and it plays a great role in
physics and engineering.
In particular, this equation is most useful for treating the
boundary value problems exhibiting certain symmetries.
Let $\kappa \geq 0$ and $\rho > 0$ be constants.
We denote by ${\mathcal C}_\kappa$ the set of all functions
$y : ( -\rho, \rho ) \to \mathbb{C}$ with the following
properties:
\begin{itemize}
\item[$(C_1)$] $y(x)$ is expressible by a power series
$\sum_{m=0}^\infty b_m x^m$ whose radius of
convergence is at least $\rho$;
\item[$(C_2)$] $\sum_{m=0}^\infty | a_m x^m | \leq \kappa
| \sum_{m=0}^\infty a_m x^m |$ for any
$x \in ( -\rho, \rho )$, where
$a_m = (m+2)(m+1)b_{m+2} -(m^2-n^2)b_m$ for all
$m \in \mathbb{N}_0$ and set $b_0 = b_1 = 0$.
\end{itemize}
In the following theorem, a local Hyers--Ulam stability of the
Chebyshev's differential equation is proved (see \cite{jr} and ref.
\cite{jk}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Theorem 2.7 %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{thm:Chebyshev}
Let $n$ be a positive integer and assume that a function
$y \in {\mathcal C}_\kappa$ satisfies the differential
inequality
$$
\big| (1-x^2) y''(x) -x y'(x) + n^2 y(x) \big| \leq \varepsilon
$$
for all $x \in (-\rho, \rho)$ and for some $\varepsilon > 0$.
Let $\rho_0 = \min \{ 1, \rho \}$.
Then there exists a Chebyshev function
$y_h : ( -\rho_0, \rho_0 ) \to \mathbb{C}$ such that
$$
| y(x) - y_h(x) |
\leq \frac{\kappa M_e \varepsilon}{2} \frac{x^2}{1-x^2}
$$
for all $x \in ( -\rho_0, \rho_0 )$, where the constant $M_e$
is defined by
$$
M_e = \max_{0 \leq i \leq \ell \leq n_e}
\frac{(2i)!}{(2\ell+1)!} (n^2-4)^{\ell-i}
$$
with $n_e = [n/\sqrt{8}]$.
\end{theorem}
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\end{thebibliography}
\end{document}
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